I started picking up stitches around the collar, and they were so loose and messy! BUT, anyway I came to the end of picking them up, and found that my working yarn was now on the inside- what am I doing wrong here? (Also, I followed a video and it showed that you do not knot the first stitch when casting on- so how do you keep it from coming apart?)

My picked up stitches are always too loose. After I’ve worked a row or more, depending on what I’m doing it may need a lot more rows, I go back to the last picked up stitch and start working out the excess yarn one stitch at a time by pulling on the legs on the knit side all the way to the beginning. I came across a handy trick that helps with this: hold contrasting crochet thread or something else thin and smooth with the working yarn when you pick up your stitches. When you work into them, treat yarn and thread as one. You can then follow the thread to easily find the legs of the stitches to adjust your tension.

Since you’re working on the RS with the yarn at the back to pick up stitches it makes sense that it’s on the inside. I don’t think you’re doing anything wrong. You’ll just move it to the other side of your work if you need to.

The first stitch won’t come apart. Do you mean that the first stitch you pick up comes loose later on? You should secure the tail end in some way: either weave it in at the beginning, or actually tie a knot (that you will undo later to weave it in) to another yarn end at that spot, if there is one.
It sometimes helps to use a size smaller needle (or two sizes?) for the picking up. Maintain good tension on your yarn! Switch to your regular needle when working on the first round after the picked up round.